Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approcimations

This paper introduces an abstract neural flow framework that unifies finite-dimensional neural networks and infinite-dimensional neural operators in a single Hilbert-space formulation. The key novelty is the separation of the continuous-depth model into two dynamics: a composition-structured flow, whose explicit Euler discretization recovers ResNet-type architectures, and a separation-structured flow, whose semi-implicit splitting discretization yields plain feedforward architectures. For piecewise-constant parameters and Leaky ReLU activations, the authors prove well-posedness and local Lipschitz dependence of the flow map, then establish what they describe as the first universal approximation theorem for flow-based models between infinite-dimensional spaces: any continuous operator O:C→Y on a compact set C in an abstract Hilbert space can be approximated arbitrarily well by R∘FθT∘P, where P and R are bounded linear maps and FθT is either flow type. They further show that the same result holds under convolutional constraints, by realizing fully connected linear dynamics exactly through constant convolution kernels. Finally, the paper transfers these continuous-time guarantees to finite-depth architectures via time discretization, proving universal approximation for both plain and ResNet-style neural operators, as well as fully connected and convolutional neural networks. The framework includes standard models such as Fourier neural operators, DeepONet, and convolutional neural operators, but its main contribution is the unified flow-based explanation of why these distinct architectures inherit universal approximation power.